Best local approximation in several variables

JOURNAL

OF APPROXIMATION

40, 343-350

THEORY

(1984)

Best Local Approximation Several Variables CHARLES

in

K. CHUI*

Department of Mathematics, Texas A & M University, College Station, Texas 77843, U.S.A HARVEY

DIAMOND

Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506. U.S.A. AND

LOUISE A. RAPHAEL* Department of Mathematics. Howard University, Washington, D.C. 20059, U.S.A. Communicated by Oved Shisha Received

January

14, 1983;

revised

May

26. 1983

I. INTRODUCTION The notion of best local approximation of a function has been introduced and developed by Chui, Shisha, and Smith (2). This notion may be briefly restated as follows. Let f: Rn + R be a function in a normed linear space X of functions with norm ]/. /I. Let I’ denote a subset of X. We wish to approximate f near a point x0 E R ” using an element of V. Let (Q, Js>0 denote a net of volumes containing x0 with diameters shrinking to zero as 6+ 0. For each 6 > 0 we “find” a ugE V which minimizes ]](S-- u)xc,]], where xos is the characteristic function of Q, . If ug converges as 6 + 0 to an element u* of V then U* is said to be the best local approximant offat x0 with respect to the net Qs. The basic underlying results in R’ were developed in (61 and [ 1, 21. These results concerned first, the best local approximant offE Cm” 10, I] from an (m + 1) dimensional subspace of P+‘[O, l]. It was found that in the * Research

partially

supported

by Army

Research

Office

Grant

DAA

G29-81-G-0011.

343 002 l-9045/84

$3.00

Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.

344

CHUI,

DIAMOND,

AND

RAPHAEL

uniform norm the best local approximant of f at x0 = 0 is obtained by choosing the element out of the subspace matching the first m derivatives off at 0 (an assumption leading to the existence and uniqueness of this element having been previously made). The so-called best local quasi-rational approximant of a function f was also considered. Here f~ Cm’ + ’ [0, I] is approximated by the ratio of functions v/w, where v E V with vc cm+yo, I] is an m dimensional subspace of functions and w E IV, with W, c Cm+“f‘[O, l] is an (n - 1) dimensional affine space in which all functions satisfy w(0) = 1. In best local quasi-rational approximation a net of functions v,/w, is generated using the pair u&, wg which for each 6 minimizes ]](wf- v)xas/]. The results obtained in one dimension using the uniform norm show that under suitable conditions onf, V, and W, , the best local quasi-rational approximant off at zero is obtained by choosing u * and w* so that the appropriate number of derivatives of w*f - U* are zero at x = 0. It was observed in [2] that this procedure for generating best local quasi-rational approximants is the same as that used to generate the Pade approximants of a function in a neighborhood of zero. In this paper we investigate best local approximation from finite dimensional subspaces and best local quasi-rational approximation of real valued functions of several variables using the L, norm 1

BEST LOCAL APPROXIMATION

345

best local quasi-rational approximant off at x = 0. This result is related to current attempts to develop a Pade approximation theory for functions of several variables.

II. NOTATION

AND DEFINITIONS

The notation and results related to analysis in R” that are used in this paper may be found in Schumaker 15, pp. 502-5081. In particular x = (x1) x2 )...) xn) is a generic point in R”, a = (a,, a2,..., a,) is an ndimensional vector (or multi-index) of nonnegative integers, D” denotes the mixed partial differentiation operator DF: DC; a.. Dz; in R”, and ]a 1= a, + ... + a,, is then the order of the derivative. The space of functions having continuous derivatives up through order m and defined on the open domain n is denoted Cm(n). We will be using the norm l]f]]r,,cu, which is [lo if(x)]” ~xJ”~ if 1

O will denote a net of measurablesubsetsof R” (referred to here as volumes) which contain the origin 0 and which have the property that for each 6, Q, contains a cube of side r6 and is contained in a cube of side 6 with 6/r, bounded as a function of 6 for all 6 > 0. Let f be a real valued function defined and continuous on some bounded open subset 0 of R ” which contains 0 and let { Q6}6>0 denote a net of volumes as described above. Let V be a finite dimensional subspace of functions continuous on fin. For each 6, let ug E V satisfy i]f- ~~]]~*,(c,)= minl,Ev Ilf- uI~~~,~~,~. If as 6- 0, ub + u* (convergence being in L,(R)) and the limit u * is independent of the choice of {Q,} then we call U* the best local approximant off at 0 from the space V with respect to the L, norm. Let f be as above and let V and W be finite dimensional subspacesof functions continuous on R. Denote by W, the affine space {w E W: w(0) = 1 }. Let {Q,} be a net of volumes. For each 6, let wg and ug satisfy

lIw8f - ~gIILp~Qs~ = minwew,,vevIIwf - 4L,cQsj.If as 6+0, uglw8 + u*/w*, and the limit is independent of the choice of (Q,}, then we call u*/w* the best local quasi-rational approximant off at 0 from V and W with respect to the L, norm. (Note that this u * has no particular relation with the V* of best local approximation above. Also, we do not require separately that ug-+ u* and wg + w*; however, in our theorem the assumptions will be such that these limits in fact do exist separately.) A vector space V of functions in C”‘(O), 0 E J2, is said to be unique!,,

346

CHLJI, DIAMOND,

AND RAPHAEL

interpolating at 0 of order m if there is a unique v E V with prescribed derivatives up through order m. If ui(x), i = l,..., k, is a basis of V, then the condition of being uniquely interpolating at zero of order m is equivalent to the Wronskian determinant being nonzero. This is the determinant of the square matrix (D’ui(0)), (al < m, i = l,..., k, where each value of a corresponds to a row of the matrix and k and m are related by k=card(a:la(
RESULTS

Our results in this section are mainly sufficient conditions for the existence of the best local approximant and the best local quasi-rational approximant for real valued functions on R”. These results hold independent of the p in L,, 1 < p < co. Basically, Taylor’s theorem gives a certain order of local approximation, if derivatives of an appropriately smooth function are matched by a function from the approximating set. Our results state that if from the approximating set, there is a unique function for which all derivatives up through order m of the error (f - v in one case or wf - u in the quasi-rational case) can be put to zero (m a nonnegative integer), then that function is the best local approximant; the net of best approximants (0, or v,/w,) generated by the net Qs of volumes converges to it on the domain n. PROPOSITION. Suppose the subspace Vc Cm+ ‘(52) has basis ui(x), i = l,..., k, and that V is uniquely interpolating at 0 of order m. If, as 6 + 0 the functions &(x) = Cf= 1ai,su,(x) satisfy \jfs IJLp(psj= 0(6~‘~‘~), then for each i, ai,s -+ 0 as 6 + 0 and&(x) + 0 in L,(0).

Proof. We will prove the result for the case of [ai,sl bounded uniformly in 6 for each i. The general case will then follow from a simple argument. We remark first that the I/. IIL,(csj norm of a function of size 0(x, cI,=m x”> is o(~J~+~‘~) as we recall that Q, is contained in a cube of side 6. The Taylor expansion offs(x) about x = 0 up through order m is given by

fs(x) = 1

Dy8(0)xa/a!

+ 0

lal
2 xa . c lCll=PTl 1

An important observation is that, as V is uniquely interpolating at 0, the ai,& are uniquely determined by the coefficients Dy8(0) via a matrix multiplication by the inverse of the matrix D%,(O) which is independent of 6. BY hypothesis,IIf, IILptas, = 46 m+ ‘lp) from which it follows that y DafsWx” a.I /I &on

= o(ijm+n’p). IiLp(Qs)

(1)

BEST

LOCAL

347

APPROXIMATION

We will show that D’&(O) + 0 as 6-+ 0 and this in turn will imply that Ui,s --$ 0. It is not difficult to show that for a polynomial in n variables of the form C, b,x’ the simple scaling transformation x = Sy results in the equation

where Q,/s = {y: 6y E Q,}. sides by ~~~~~~~ we obtain \‘ II i=Ln

Applying

P-Vgcs(0) a. I

this result to (1) and dividing

ya

= o(1). IILBtQdS)

both

(2)

We will prove that b,, + 0 by showing Redefine b,, = (d'"'-"D"f,(O)/a!). that every sequence (b,,J with ai + 0 has a subsequence{ba,,i ) such that J b,,si.+ 0. Recall that for each 6, Q, is contained within a cube of side 6 and contains a cube, denoted by I,, which has sides at least as large as cd, with c > 0 a fixed constant. It follows then that Q,/6 is contained for all 6 within the cube ~7 I-1, 1] and that the cubes 1,/J have sides of length at least c. Now the centers of the cubes (Zs,/Ji} all lie within the cube X: [-1, 1] so that there is a subsequenceof {Si}, {Si,} = {ek), such that the centers of the cubes Z,,/E~ converge. This in turn implies that for large k, the cubes I,/&, have substantial intersection. Indeed, there exists a K such that nkaK (I,/&,) contains a cube B of side c/4 and then B c Q,/sk for k > K. We then have

where the last equality follows from (2). This implies finally that bqQ + 0 SO that we have proved that b, s + 0 as 6 + 0. From the definition of b,, we then obtain P&(O) = o(P-‘“I) as 6 + 0, so that in particular DQc,(O)+ 0. This completes the proof in the case that the u~,~, the components of&(x) with respect to the basis (ui(x)} are uniformly bounded in 6. Suppose that the ai,s were not bounded in 6. Then we would have maxi ]Ui 6.]+ co for a sequence Sj -9 0. If we then define g,,(x) = [max, (cz~,~~]] “‘J,(x), then m+n’p) and the components of g,) with respect to the basis IIgSjllL,(Qaj) = OCsj {ui} are uniformly bounded in j. The previous arguments then show that g,. + 0 asj + co. But one of the components of g,, must have absolute value 1 for each j and a contradiction is obtained. This completes the proof of the proposition.

348

CHLJI,DIAMOND,AND

RAPHAEL

THEOREM 1. Suppose the subspace V c Cm+‘(Q) is uniquely interpolating at 0 of order m. Then the best local approximant off E Cm+’ at 0 from V is the unique v E V whose derivatives up through order m match those off at 0.

Proof: By choosing v* E V so as to match the derivatives up through order m of f at 0, we are guaranteed by Taylor’s theorem that minvev 11 f - v IIL,(cs)< I(f - v* (IL,tas,= o(B~+~‘~). If we write us, the best L, approximants of f on Q,, in the form vg = v* + us’ we then have 46 m+n’p)> Ilf - &,(Q,) = Ilf - u* - 4 IIL,(Qa) > II4 III&) “‘+“lp). But then the proposition llf - ZJ* IILpcQ,) I so that II4 lILofQs~ = 46 implies that 11 vt II-+ 0 so that vg + v’ and the theorem is proved. Next we consider the best quasi-rational approximant ofJ We let V and W denote finite dimensional subspacesof Cm”(n), we set W, = {w E W: w(0) = 0) and W, = {w E W: w(0) = 1}. THEOREM 2. Let V and W,, be subspacesof functions in C”“(Q) as defined above and let f E Cm+‘(Q). Suppose the subspacesfW, and V are disjoint except for 0. If the vector space { fw - v: w E W,,, v E V} is uniquely interpolating at 0 of order m, then the best local quasi-rational approximant off at 0 is obtained by choosing the unique v * E V and w* E W, so that all the derivatives of fw * - v* up through order m are zero.

Proof First observe that if w1 is a function in W with w,(O) = 1, then we can express W, as W, = w, + W,. In that case fw - v with w E W, and v E V may be written asfw, + ( fwO - v) with w, E W,. The set of functions in parentheses was assumed uniquely interpolating up through order m so that there is indeed a unique w$ E W, and v* E V which makesfw - v have all zero derivatives up through order m. (Nonuniqueness would imply the existence of a w0 and v such that fwO - v E 0, which contradicts the disjointness assumption in the statement of the theorem.) If wO,&and vg denote solutions of minw,Ew,,vE,, IIfw, + fwO - vllLp(ea,, then an argument similar to the one in Theorem 1 shows that fw,,, - vg -fw,* - v*. The proposition shows, moreover, that if we write out the components of fW0,6 - vg with respect to bases of fW,, and V, then we may conclude separately that w~,~+ w$ and vg + v* and finally, that wg = w, + w~.~--f w* = w, + w$. This proves the theorem. As a method of approximation of a function of several variables by polynomials in the quasi-rational sense,interpolation of derivatives arises as the central technique in the theory of Padi approximants. While the one variable theory is well developed, in several variables ambiguity arises in the choice of which derivatives should be matched when, as is usually the case, all derivatives up to order m - 1, say, can be matched but not all of those up

BEST

LOCAL

349

APPROXIMATION

to order m. See, for instance, Lutterodt [4] and Karllson and Wallin [3]. From the viewpoint of the best local approximant, however, it does not even make sense to consider a quasi-rational approximant unless the approximating space is uniquely interpolating of order m. This will be illustrated by an example. Suppose we wish a best local approximant of the function xy using the vector space spanned by 1, x, y, x2 + xy + y*. The vector space uniquely interpolates any specification of one second-order derivative and all derivatives up through the first. We will see, however, that the net of approximations generated by the volumes Q, depends sensitively on the dimensions of the volumes. Writing out explicitly the minimization problem, we have

minllf- UllI.,~Q,) .(dIxy-a,-a,x-a,y-a,(x*+x~+?:*)]“dxd~~ = min ijC6 ,o

IlP

I

0

’

where we are using for Q, a net of rectangles with dimensions 6 x c6 (c < 1). We now perform the scaling transformations x = as, y = c dt, a, = #y,, a, = 67,) a2 = &I,, a3 = yj, and obtain

=

a2 + */P min

l/P

[cst - y. - 7,s - yzct - yJs* + cst + c2t2)lp ds dt

.

We see that the optimal choices of yo, y1 , y2, and y3 are independent of 6, so that in particular, the optimal choice of a3 is independent of 6. However, it clearly depends on the choice of the constant c, so that, although for each fixed c the approximations ug tend to a limit, this limit is not independent of c. This means, of course, that the limit is not the same for all nets Q, of rectangles and thus a best local approximant does not exist. This example clearly generalizes to more complicated situations; whenever the number of basis functions is such that interpolation up through order m - 1 is possible, but not all the way up to order m, then the limit of the net of best approximants ug will depend on the dimensions of the net of volumes Q, (see Note added in proof). Note analysis

added in proof: This dependence of multipoint local approximation.

is treated

in some

detail

in 17 1 as part

of an

350

CHUI,

DIAMOND,

AND

RAPHAEL

REFERENCES

1. C. K. CHUI, 0. SHISHA, AND P. W. SMITH, Padit approximants as limits of best rational approximants, J. Approx. Theory 12 (1974), 201-204. 2. C. K. CHUI, 0. SHISHA, AND P. W. SMITH, Best local approximation, J. Approx. Theory 15 (1975), 371-381. 3. J. KARLSSON AND H. WALLIN, Rational approximation by an interpolation procedure in several variables, in “Pad& and Rational Approximation” (E. B. Saff and R. S. Varga. Eds.), Academic Press, New York, 1977. 4. C. H. LUTTERODT, On convergence of (.u,v)-sequences of univalent rational approximants to meromorphic functions in C”, preprint. 5. L. L. SCHLJMAKER,“Spline Functions: Basic Theory,” Wiley, New York, 1981. 6. J. L. WALSH, Padtr approximants as limits of rational functions of best approximation, real domain, J. Approx. Theory 11 (1974), 225-230. 7. C. K. CHUI, H. DIAMOND, AND L. A. RAPHAEL, On best data approximation, J. Approx. Theory and Its Appls, to appear.